{"id":4354,"date":"2025-07-30T22:25:15","date_gmt":"2025-07-30T13:25:15","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/?p=4354"},"modified":"2025-07-30T22:25:17","modified_gmt":"2025-07-30T13:25:17","slug":"robust-estimation-in-regression-analysis-building-models-resilient-to-outliers","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/robust-estimation-in-regression-analysis-building-models-resilient-to-outliers\/","title":{"rendered":"\u56de\u5e30\u5206\u6790\u306b\u304a\u3051\u308b\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\uff1a\u5916\u308c\u5024\u306b\u5f37\u3044\u30e2\u30c7\u30eb\u3092\u69cb\u7bc9\u3059\u308b"},"content":{"rendered":"\n<p>\u56de\u5e30\u5206\u6790\u306f\u3001\u5909\u6570\u9593\u306e\u95a2\u4fc2\u6027\u3092\u660e\u3089\u304b\u306b\u3059\u308b\u5f37\u529b\u306a\u7d71\u8a08\u30c4\u30fc\u30eb\u3067\u3042\u308b\u3002\u3057\u304b\u3057\u3001\u30c7\u30fc\u30bf\u306e\u4e2d\u306b\u300c<strong>\u5916\u308c\u5024<\/strong>\u300d\u3068\u547c\u3070\u308c\u308b\u7279\u7570\u306a\u5024\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u3001\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3067\u306f\u5206\u6790\u7d50\u679c\u304c\u5927\u304d\u304f\u6b6a\u3081\u3089\u308c\u3066\u3057\u307e\u3046\u3053\u3068\u304c\u3042\u308b\u3002\u3053\u306e\u3088\u3046\u306a\u554f\u984c\u3092\u89e3\u6c7a\u3057\u3001\u3088\u308a\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u30e2\u30c7\u30eb\u3092\u69cb\u7bc9\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u308b\u306e\u304c\u300c<strong>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5<\/strong>\u300d\u3067\u3042\u308b\u3002\u672c\u8a18\u4e8b\u3067\u306f\u3001\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u306e\u6982\u5ff5\u3001\u5fc5\u8981\u6027\u3001\u5177\u4f53\u7684\u306a\u624b\u6cd5\u3001\u305d\u3057\u3066R\u3067\u306e\u8a08\u7b97\u4f8b\u3092\u901a\u3058\u3066\u3001\u305d\u306e\u6709\u7528\u6027\u3092\u89e3\u8aac\u3059\u308b\u3002<\/p>\n\n\n\n<!--more-->\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u3068\u306f<\/h2>\n\n\n\n<p>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\uff08Robust Estimation\uff09\u3068\u306f\u3001\u30c7\u30fc\u30bf\u4e2d\u306b\u5916\u308c\u5024\u3084\u8aa4\u5dee\u306e\u5206\u5e03\u304c\u6b63\u898f\u5206\u5e03\u304b\u3089\u9038\u8131\u3057\u3066\u3044\u308b\u5834\u5408\u3067\u3082\u3001\u63a8\u5b9a\u7d50\u679c\u304c\u5927\u304d\u304f\u5f71\u97ff\u3092\u53d7\u3051\u306b\u304f\u3044\u7d71\u8a08\u7684\u63a8\u5b9a\u65b9\u6cd5\u306e\u7dcf\u79f0\u3067\u3042\u308b\u3002\u4e00\u822c\u7684\u306a\u56de\u5e30\u5206\u6790\u3067\u7528\u3044\u3089\u308c\u308b<strong>\u6700\u5c0f\u4e8c\u4e57\u6cd5<\/strong>\uff08Ordinary Least Squares; OLS\uff09\u306f\u3001\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u4fc2\u6570\u3092\u63a8\u5b9a\u3059\u308b\u304c\u3001\u3053\u306e\u65b9\u6cd5\u306f\u5916\u308c\u5024\u306b\u5bfe\u3057\u3066\u975e\u5e38\u306b\u654f\u611f\u3067\u3042\u308b\u3002\u5916\u308c\u5024\u306f\u6b8b\u5dee\u3092\u5927\u304d\u304f\u3059\u308b\u305f\u3081\u3001\u305d\u306e\u4e8c\u4e57\u306f\u3055\u3089\u306b\u5927\u304d\u304f\u306a\u308a\u3001\u30e2\u30c7\u30eb\u304c\u5916\u308c\u5024\u306b\u5f37\u304f\u5f15\u304d\u305a\u3089\u308c\u3066\u3057\u307e\u3046\u50be\u5411\u304c\u3042\u308b\u3002<\/p>\n\n\n\n<p>\u3053\u308c\u306b\u5bfe\u3057\u3001\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306f\u3001\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u5c0f\u3055\u304f\u3059\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u308b\u3002\u4f8b\u3048\u3070\u3001\u6b8b\u5dee\u306e\u7d76\u5bfe\u5024\u3092\u4f7f\u7528\u3057\u305f\u308a\u3001\u5916\u308c\u5024\u306b\u5c0f\u3055\u306a\u91cd\u307f\u3092\u4e0e\u3048\u305f\u308a\u3059\u308b\u3053\u3068\u3067\u3001\u5916\u308c\u5024\u304c\u63a8\u5b9a\u7d50\u679c\u306b\u4e0e\u3048\u308b\u5f71\u97ff\u3092\u6291\u5236\u3059\u308b\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u304c\u5fc5\u8981\u306a\u5834\u5408<\/h2>\n\n\n\n<p>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u304c\u5fc5\u8981\u3068\u306a\u308b\u4e3b\u306a\u30b1\u30fc\u30b9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u5916\u308c\u5024\u306e\u5b58\u5728<\/strong>: \u305d\u306e\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306b\u304a\u3044\u3066\u975e\u5e38\u306b\u7a00\u306a\u89b3\u6e2c\u5024\u3068\u3044\u3046\u5f62\u3067\u5916\u308c\u5024\u306f\u767a\u751f\u3059\u308b\u3002\u3053\u308c\u3089\u306e\u5916\u308c\u5024\u304c\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u63a8\u5b9a\u3092\u6b6a\u3081\u308b\u53ef\u80fd\u6027\u304c\u9ad8\u3044\u5834\u5408\u3001\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u304c\u6709\u52b9\u3067\u3042\u308b\u3002<\/li>\n\n\n\n<li><strong>\u8aa4\u5dee\u306e\u975e\u6b63\u898f\u6027<\/strong>: \u6700\u5c0f\u4e8c\u4e57\u6cd5\u306f\u3001\u8aa4\u5dee\u9805\u304c\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u3068\u3044\u3046\u4eee\u5b9a\u306e\u3082\u3068\u3067\u6700\u3082\u52b9\u7387\u7684\u3067\u3042\u308b\u3068\u3055\u308c\u3066\u3044\u308b\u3002\u3057\u304b\u3057\u3001\u5b9f\u969b\u306e\u30c7\u30fc\u30bf\u3067\u306f\u8aa4\u5dee\u304c\u6b63\u898f\u5206\u5e03\u304b\u3089\u9038\u8131\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u5c11\u306a\u304f\u306a\u3044\u3002\u3053\u306e\u3088\u3046\u306a\u5834\u5408\u3067\u3082\u3001\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u306f\u6bd4\u8f03\u7684\u5b89\u5b9a\u3057\u305f\u7d50\u679c\u3092\u63d0\u4f9b\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u30e2\u30c7\u30eb\u306e\u9811\u5065\u6027\u5411\u4e0a<\/strong>: \u63a8\u5b9a\u3055\u308c\u305f\u30e2\u30c7\u30eb\u304c\u3001\u672a\u77e5\u306e\u30c7\u30fc\u30bf\u3084\u5c06\u6765\u306e\u30c7\u30fc\u30bf\u306b\u5bfe\u3057\u3066\u3082\u5b89\u5b9a\u3057\u305f\u4e88\u6e2c\u6027\u80fd\u3092\u6301\u3064\u3053\u3068\u3092\u91cd\u8996\u3059\u308b\u5834\u5408\u3001\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u53d7\u3051\u306b\u304f\u3044\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u306f\u6709\u52b9\u306a\u9078\u629e\u80a2\u3068\u306a\u308b\u3002<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div id=\"biost-3325584588\" class=\"biost- biost-entity-placement\"><p style=\"text-align: center;\"><span style=\"font-size: 20px;\"><strong><a href=\"https:\/\/best-biostatistics.com\/kmhl\">\uff1e\uff1e\u3082\u3046\u7d71\u8a08\u3067\u60a9\u3080\u306e\u306f\u7d42\u308f\u308a\u306b\u3057\u307e\u305b\u3093\u304b\uff1f\u00a0<\/a><\/strong><\/span><\/p>\r\n<a href=\"https:\/\/best-biostatistics.com\/kmhl\"><img class=\"aligncenter wp-image-2794 size-full\" src=\"https:\/\/best-biostatistics.com\/wp\/wp-content\/uploads\/2023\/11\/bn_r_03.png\" alt=\"\" width=\"500\" height=\"327\" \/><\/a>\r\n<p style=\"text-align: center;\"><span style=\"color: #ff0000; font-size: 20px;\"><strong><span class=\"marker2\">\u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div><h2 class=\"wp-block-heading\">\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u306e\u5177\u4f53\u4f8b<\/h2>\n\n\n\n<p>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u306b\u306f\u3044\u304f\u3064\u304b\u306e\u624b\u6cd5\u304c\u3042\u308b\u304c\u3001\u4ee3\u8868\u7684\u306a\u3082\u306e\u3068\u3057\u3066\u4ee5\u4e0b\u304c\u6319\u3052\u3089\u308c\u308b\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>M\u63a8\u5b9a (M-estimation)<\/strong>: \u6700\u5c0f\u4e8c\u4e57\u6cd5\u304c\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u3092\u6700\u5c0f\u5316\u3059\u308b\u306e\u306b\u5bfe\u3057\u3001M\u63a8\u5b9a\u306f\u6b8b\u5dee\u306e\u4f55\u3089\u304b\u306e\u95a2\u6570$ \\rho(\\cdot) $\u3092\u6700\u5c0f\u5316\u3059\u308b\u3002\u3053\u306e$ \\rho(\\cdot) $\u306f\u3001\u5916\u308c\u5024\u306b\u5bfe\u3057\u3066\u6b8b\u5dee\u304c\u5927\u304d\u304f\u306a\u3063\u3066\u3082\u3001\u305d\u306e\u5f71\u97ff\u3092\u6291\u3048\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u308b\u3002\u30d2\u30e5\u30fc\u30d0\u30fc\u95a2\u6570\uff08Huber function\uff09\u3084\u30c1\u30e5\u30fc\u30ad\u30fc\u95a2\u6570\uff08Tukey&#8217;s biweight function\uff09\u306a\u3069\u304c\u3088\u304f\u7528\u3044\u3089\u308c\u308b\u3002\n<ul class=\"wp-block-list\">\n<li><strong>\u30d2\u30e5\u30fc\u30d0\u30fc\u95a2\u6570<\/strong>: \u3042\u308b\u95be\u5024\u3092\u8d85\u3048\u308b\u3068\u3001\u6b8b\u5dee\u306e\u4e8c\u4e57\u3067\u306f\u306a\u304f\u7d76\u5bfe\u5024\u306b\u6bd4\u4f8b\u3059\u308b\u3088\u3046\u306b\u306a\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u6975\u7aef\u306a\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u6291\u5236\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u30c1\u30e5\u30fc\u30ad\u30fc\u95a2\u6570<\/strong>: \u3042\u308b\u95be\u5024\u3092\u8d85\u3048\u308b\u3068\u3001\u6b8b\u5dee\u306e\u5f71\u97ff\u3092\u5b8c\u5168\u306b\u7121\u8996\u3059\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u308b\u3002\u3088\u308a\u5f37\u529b\u306b\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u6392\u9664\u3067\u304d\u308b\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>S\u63a8\u5b9a (S-estimation)<\/strong>: \u6b8b\u5dee\u306e\u5c3a\u5ea6\uff08\u4f8b\uff1a\u6b8b\u5dee\u306e\u5206\u6563\uff09\u3092\u6700\u5c0f\u5316\u3059\u308b\u63a8\u5b9a\u6cd5\u3067\u3042\u308b\u3002M\u63a8\u5b9a\u3088\u308a\u3082\u3055\u3089\u306b\u9ad8\u3044\u5916\u308c\u5024\u8010\u6027\u3092\u6301\u3064\u3053\u3068\u304c\u7279\u5fb4\u3067\u3042\u308b\u3002<\/li>\n\n\n\n<li><strong>MM\u63a8\u5b9a (MM-estimation)<\/strong>: M\u63a8\u5b9a\u3068S\u63a8\u5b9a\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u65b9\u6cd5\u3067\u3001\u9ad8\u3044\u5916\u308c\u5024\u8010\u6027\u3068\u52b9\u7387\u6027\u3092\u4e21\u7acb\u3055\u305b\u308b\u3002\u901a\u5e38\u3001\u307e\u305aS\u63a8\u5b9a\u3067\u521d\u671f\u5024\u3092\u6c42\u3081\u3001\u305d\u306e\u521d\u671f\u5024\u3092\u7528\u3044\u3066M\u63a8\u5b9a\u3092\u884c\u3046\u3053\u3068\u3067\u3001\u3088\u308a\u52b9\u7387\u7684\u306a\u63a8\u5b9a\u3092\u5b9f\u73fe\u3059\u308b\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8a18\u4e8b\u672b\u306b\uff13\u3064\u306e\u63a8\u5b9a\u6cd5\u306e\u8a73\u7d30\u3092\u63d0\u793a\u3057\u3066\u3042\u308b\u306e\u3067\u3001\u8208\u5473\u304c\u3042\u308c\u3070\u53c2\u7167\u3057\u3066\u307b\u3057\u3044\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">R\u3067\u306e\u8a08\u7b97\u4f8b<\/h2>\n\n\n\n<p>R\u3067\u306f\u3001<code>MASS<\/code>\u30d1\u30c3\u30b1\u30fc\u30b8\u306e<code>rlm()<\/code>\u95a2\u6570\u3084\u3001<code>robustbase<\/code>\u30d1\u30c3\u30b1\u30fc\u30b8\u306e<code>lmrob()<\/code>\u95a2\u6570\u306a\u3069\u3092\u7528\u3044\u3066\u30ed\u30d0\u30b9\u30c8\u56de\u5e30\u3092\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067\u306f<code>MASS<\/code>\u30d1\u30c3\u30b1\u30fc\u30b8\u306e<code>rlm()<\/code>\u95a2\u6570\u3092\u7528\u3044\u305f\u4f8b\u3092\u793a\u3059\u3002<\/p>\n\n\n\n<p>R \u30b9\u30af\u30ea\u30d7\u30c8\u4f8b\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code># \u30c7\u30fc\u30bf\u306e\u751f\u6210\nset.seed(123)\nx &lt;- 1:20\ny &lt;- 2 * x + rnorm(20, 0, 2)\n\n# \u5916\u308c\u5024\u3092\u5c0e\u5165\ny&#91;c(5, 15)] &lt;- c(50, 1) # 2\u3064\u306e\u5916\u308c\u5024\u3092\u610f\u56f3\u7684\u306b\u5c0e\u5165\n\n# \u30c7\u30fc\u30bf\u30d5\u30ec\u30fc\u30e0\u306b\u307e\u3068\u3081\u308b\ndata_df &lt;- data.frame(x = x, y = y)\n\n# \u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308b\u56de\u5e30\nlm_model &lt;- lm(y ~ x, data = data_df)\nsummary(lm_model)\n\n# \u30ed\u30d0\u30b9\u30c8\u56de\u5e30\uff08M\u63a8\u5b9a - Huber loss\uff09\nlibrary(MASS)\nrlm_model_huber &lt;- rlm(y ~ x, data = data_df, psi = psi.huber)\nsummary(rlm_model_huber)\n\n# \u30ed\u30d0\u30b9\u30c8\u56de\u5e30\uff08M\u63a8\u5b9a - Tukey's biweight loss\uff09\nrlm_model_tukey &lt;- rlm(y ~ x, data = data_df, psi = psi.bisquare)\nsummary(rlm_model_tukey)\n\n# \u7d50\u679c\u306e\u6bd4\u8f03\u30d7\u30ed\u30c3\u30c8\nlibrary(ggplot2)\nggplot(data_df, aes(x = x, y = y)) +\n  geom_point() +\n  geom_abline(aes(intercept = lm_model$coefficients&#91;1], slope = lm_model$coefficients&#91;2], color = \"\u6700\u5c0f\u4e8c\u4e57\u6cd5\", linetype = \"\u6700\u5c0f\u4e8c\u4e57\u6cd5\")) +\n  geom_abline(aes(intercept = rlm_model_huber$coefficients&#91;1], slope = rlm_model_huber$coefficients&#91;2], color = \"Huber\", linetype = \"Huber\")) +\n  geom_abline(aes(intercept = rlm_model_tukey$coefficients&#91;1], slope = rlm_model_tukey$coefficients&#91;2], color = \"Tukey\", linetype = \"Tukey\")) +\n  scale_color_manual(name = \"\u30e2\u30c7\u30eb\", \n                    values = c(\"\u6700\u5c0f\u4e8c\u4e57\u6cd5\" = \"red\", \"Huber\" = \"blue\", \"Tukey\" = \"darkgreen\")) +\n  scale_linetype_manual(name = \"\u30e2\u30c7\u30eb\",\n                       values = c(\"\u6700\u5c0f\u4e8c\u4e57\u6cd5\" = \"dashed\", \"Huber\" = \"solid\", \"Tukey\" = \"solid\")) +\n  labs(title = \"\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u6bd4\u8f03\", x = \"x\", y = \"y\") +\n  theme_minimal()\nggsave(\"robust_estimation_linear_reg.png\", width = 8, height = 6)\n<\/code><\/pre>\n\n\n\n<p>\u3053\u306e\u30b3\u30fc\u30c9\u3092\u5b9f\u884c\u3059\u308b\u3068\u3001\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308b\u56de\u5e30\u76f4\u7dda\u304c\u5916\u308c\u5024\u306b\u5f37\u304f\u5f15\u3063\u5f35\u3089\u308c\u3066\u3044\u308b\u306e\u306b\u5bfe\u3057\u3001\u30ed\u30d0\u30b9\u30c8\u56de\u5e30\u306b\u3088\u308b\u76f4\u7dda\u306f\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u53d7\u3051\u306b\u304f\u304f\u3001\u30c7\u30fc\u30bf\u306e\u672c\u6765\u306e\u50be\u5411\u306b\u8fd1\u3044\u5f62\u3067\u30d5\u30a3\u30c3\u30c8\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u8996\u899a\u7684\u306b\u3082\u78ba\u8a8d\u3067\u304d\u308b\u3002\u6700\u5c0f\u4e8c\u4e57\u6cd5\uff08\u8d64\u7834\u7dda\uff09\u306f\u3001\u5916\u308c\u5024\u306b\u5f71\u97ff\u3092\u53d7\u3051\u3066\u3001\u672c\u6765\u306e\u50be\u304d\u3088\u308a\u5c0f\u3055\u3044\u50be\u304d\u306b\u306a\u3063\u3066\u3044\u308b\uff08\u4e0b\u56f3\uff09<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1024\" height=\"768\" src=\"https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2025\/07\/robust_estimation_linear_reg-1024x768.png\" alt=\"\" class=\"wp-image-4355\" srcset=\"https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2025\/07\/robust_estimation_linear_reg-1024x768.png 1024w, https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2025\/07\/robust_estimation_linear_reg-300x225.png 300w, https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2025\/07\/robust_estimation_linear_reg-768x576.png 768w, https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2025\/07\/robust_estimation_linear_reg-1536x1152.png 1536w, https:\/\/best-biostatistics.com\/toukei-er\/wp-content\/uploads\/2025\/07\/robust_estimation_linear_reg-2048x1536.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">\u7d50\u679c\u89e3\u91c8\u4f8b<\/h3>\n\n\n\n<p>\u4e0a\u8a18\u306eR\u30b3\u30fc\u30c9\u306e\u51fa\u529b\u4f8b\u3092\u53c2\u8003\u306b\u3001\u7d50\u679c\u306e\u89e3\u91c8\u3092\u884c\u3046\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u6700\u5c0f\u4e8c\u4e57\u6cd5\uff08OLS\uff09\u306e\u7d50\u679c<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>> # \u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308b\u56de\u5e30\n> lm_model &lt;- lm(y ~ x, data = data_df)\n> summary(lm_model)\n\nCall:\nlm(formula = y ~ x, data = data_df)\n\nResiduals:\n    Min      1Q  Median      3Q     Max \n-27.403  -3.382  -0.303   2.235  36.102 \n\nCoefficients:\n            Estimate Std. Error t value Pr(>|t|)   \n(Intercept)   6.6456     5.2477   1.266  0.22152   \nx             1.4505     0.4381   3.311  0.00388 **\n---\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\n\nResidual standard error: 11.3 on 18 degrees of freedom\nMultiple R-squared:  0.3785,    Adjusted R-squared:  0.344 \nF-statistic: 10.96 on 1 and 18 DF,  p-value: 0.003884\n\n> <\/code><\/pre>\n\n\n\n<p>\u5916\u308c\u5024\u306e\u5f71\u97ff\u306b\u3088\u308a\u3001\u5207\u7247\u306f\u9ad8\u3081\u306b\u3001<code>x<\/code>\u306e\u4fc2\u6570\uff08\u771f\u306e\u5024\u306f2\uff09\u306f\u4f4e\u3081\uff08\u7d041.45\uff09\u306b\u63a8\u5b9a\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u7279\u306b\u5916\u308c\u502450\u306e\u5f71\u97ff\u304c\u5f37\u304f\u51fa\u3066\u3044\u308b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u30ed\u30d0\u30b9\u30c8\u56de\u5e30\uff08Huber M\u63a8\u5b9a\uff09\u306e\u7d50\u679c<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>> # \u30ed\u30d0\u30b9\u30c8\u56de\u5e30\uff08M\u63a8\u5b9a - Huber loss\uff09\n> library(MASS)\n> rlm_model_huber &lt;- rlm(y ~ x, data = data_df, psi = psi.huber)\n> summary(rlm_model_huber)\n\nCall: rlm(formula = y ~ x, data = data_df, psi = psi.huber)\nResiduals:\n     Min       1Q   Median       3Q      Max \n-28.9935  -1.6168   0.2043   1.8367  39.1758 \n\nCoefficients:\n            Value   Std. Error t value\n(Intercept)  1.2396  1.3791     0.8989\nx            1.9169  0.1151    16.6512\n\nResidual standard error: 2.675 on 18 degrees of freedom\n> <\/code><\/pre>\n\n\n\n<p>Huber M\u63a8\u5b9a\u3092\u7528\u3044\u305f\u5834\u5408\u3001\u5207\u7247\u306f\u7d041.24\u3001<code>x<\/code>\u306e\u4fc2\u6570\u306f\u7d041.92\u3068\u63a8\u5b9a\u3055\u308c\u3066\u3044\u308b\u3002\u3053\u308c\u306f\u3001\u5916\u308c\u5024\u304c\u5c0e\u5165\u3055\u308c\u308b\u524d\u306e\u771f\u306e\u4fc2\u6570\uff08\u5207\u72470\u3001\u4fc2\u65702\uff09\u306b\u8fd1\u3044\u5024\u3067\u3042\u308a\u3001\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u52b9\u679c\u7684\u306b\u6291\u5236\u3067\u304d\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u30ed\u30d0\u30b9\u30c8\u56de\u5e30\uff08Tukey&#8217;s biweight M\u63a8\u5b9a\uff09\u306e\u7d50\u679c<\/h4>\n\n\n\n<pre class=\"wp-block-code\"><code>> # \u30ed\u30d0\u30b9\u30c8\u56de\u5e30\uff08M\u63a8\u5b9a - Tukey's biweight loss\uff09\n> rlm_model_tukey &lt;- rlm(y ~ x, data = data_df, psi = psi.bisquare)\n> summary(rlm_model_tukey)\n\nCall: rlm(formula = y ~ x, data = data_df, psi = psi.bisquare)\nResiduals:\n     Min       1Q   Median       3Q      Max \n-29.3463  -1.3606   0.1061   1.3358  39.5633 \n\nCoefficients:\n            Value   Std. Error t value\n(Intercept)  0.4818  1.0859     0.4437\nx            1.9910  0.0907    21.9627\n\nResidual standard error: 2.133 on 18 degrees of freedom\n> <\/code><\/pre>\n\n\n\n<p>Tukey&#8217;s biweight M\u63a8\u5b9a\u3092\u7528\u3044\u305f\u5834\u5408\u3082\u540c\u69d8\u306b\u3001\u5207\u7247\u306f\u7d040.48\u3001<code>x<\/code>\u306e\u4fc2\u6570\u306f\u7d041.99\u3068\u63a8\u5b9a\u3055\u308c\u3066\u304a\u308a\u3001Huber\u63a8\u5b9a\u3088\u308a\u3082\u3055\u3089\u306b\u771f\u306e\u5024\u306b\u8fd1\u3044\u7d50\u679c\u304c\u5f97\u3089\u308c\u3066\u3044\u308b\u3002\u3053\u308c\u306f\u3001Tukey&#8217;s biweight\u304c\u6975\u7aef\u306a\u5916\u308c\u5024\u306b\u5bfe\u3057\u3066\u3088\u308a\u5f37\u3044\u5f71\u97ff\u529b\u6291\u5236\u52b9\u679c\u3092\u6301\u3064\u305f\u3081\u3068\u8003\u3048\u3089\u308c\u308b\u3002<\/p>\n\n\n\n<p>\u3053\u308c\u3089\u306e\u7d50\u679c\u304b\u3089\u3001\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3001\u5916\u308c\u5024\u304c\u5b58\u5728\u3059\u308b\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u306b\u304a\u3044\u3066\u3082\u3001\u3088\u308a\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u56de\u5e30\u4fc2\u6570\u3092\u63a8\u5b9a\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u307e\u3068\u3081<\/h2>\n\n\n\n<p>\u56de\u5e30\u5206\u6790\u306b\u304a\u3044\u3066\u5916\u308c\u5024\u306f\u5e38\u306b\u554f\u984c\u3068\u306a\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u3002\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3067\u306f\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u5f37\u304f\u53d7\u3051\u3066\u3057\u307e\u3044\u3001\u5206\u6790\u7d50\u679c\u304c\u6b6a\u3081\u3089\u308c\u308b\u30ea\u30b9\u30af\u304c\u3042\u308b\u3002\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306f\u3001\u3053\u306e\u3088\u3046\u306a\u5916\u308c\u5024\u304c\u5b58\u5728\u3059\u308b\u72b6\u6cc1\u4e0b\u3067\u3082\u3001\u3088\u308a\u5b89\u5b9a\u3057\u305f\u4fe1\u983c\u6027\u306e\u9ad8\u3044\u56de\u5e30\u4fc2\u6570\u3092\u63a8\u5b9a\u3059\u308b\u305f\u3081\u306e\u5f37\u529b\u306a\u30c4\u30fc\u30eb\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<p>M\u63a8\u5b9a\u3001S\u63a8\u5b9a\u3001MM\u63a8\u5b9a\u306a\u3069\u3001\u69d8\u3005\u306a\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u304c\u5b58\u5728\u3057\u3001\u305d\u308c\u305e\u308c\u7570\u306a\u308b\u7279\u6027\u3092\u6301\u3064\u3002\u30c7\u30fc\u30bf\u306e\u6027\u8cea\u3084\u5916\u308c\u5024\u306e\u30bf\u30a4\u30d7\u306b\u5fdc\u3058\u3066\u9069\u5207\u306a\u624b\u6cd5\u3092\u9078\u629e\u3059\u308b\u3053\u3068\u304c\u91cd\u8981\u3067\u3042\u308b\u3002R\u306a\u3069\u306e\u7d71\u8a08\u30bd\u30d5\u30c8\u30a6\u30a7\u30a2\u3092\u6d3b\u7528\u3059\u308b\u3053\u3068\u3067\u3001\u3053\u308c\u3089\u306e\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u3092\u5bb9\u6613\u306b\u9069\u7528\u3057\u3001\u5916\u308c\u5024\u306b\u5f37\u3044\u56de\u5e30\u30e2\u30c7\u30eb\u3092\u69cb\u7bc9\u3067\u304d\u308b\u3002\u30c7\u30fc\u30bf\u5206\u6790\u3092\u884c\u3046\u969b\u306b\u306f\u3001\u5916\u308c\u5024\u306e\u6709\u7121\u3092\u5e38\u306b\u610f\u8b58\u3057\u3001\u5fc5\u8981\u306b\u5fdc\u3058\u3066\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306e\u5c0e\u5165\u3092\u691c\u8a0e\u3059\u308b\u3053\u3068\u3067\u3001\u3088\u308a\u6b63\u78ba\u3067\u9811\u5065\u306a\u77e5\u898b\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3060\u308d\u3046\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u88dc\u8db3\uff1a\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u306e\u8a73\u7d30<\/h2>\n\n\n\n<p>\u627f\u77e5\u3057\u305f\u3002\u56de\u5e30\u5206\u6790\u306b\u304a\u3051\u308b\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u3001M\u63a8\u5b9a\u3001S\u63a8\u5b9a\u3001MM\u63a8\u5b9a\u306b\u3064\u3044\u3066\u3001\u305d\u308c\u305e\u308c\u306e\u7279\u5fb4\u3068\u4ed5\u7d44\u307f\u3092\u8a73\u3057\u304f\u89e3\u8aac\u3059\u308b\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\uff08Robust Estimation\uff09\u3068\u306f<\/h3>\n\n\n\n<p>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306f\u3001\u30c7\u30fc\u30bf\u4e2d\u306b\u5b58\u5728\u3059\u308b<strong>\u5916\u308c\u5024\uff08outliers\uff09<\/strong>\u3084\u3001\u8aa4\u5dee\u304c\u6b63\u898f\u5206\u5e03\u306b\u5f93\u308f\u306a\u3044\uff08<strong>\u975e\u6b63\u898f\u6027<\/strong>\uff09\u5834\u5408\u306b\u3001\u63a8\u5b9a\u7d50\u679c\u304c\u305d\u308c\u3089\u306e\u5f71\u97ff\u3092\u5927\u304d\u304f\u53d7\u3051\u306a\u3044\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u305f\u7d71\u8a08\u7684\u63a8\u5b9a\u65b9\u6cd5\u306e\u7dcf\u79f0\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<p>\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\uff08OLS: Ordinary Least Squares\uff09\u306f\u3001\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u56de\u5e30\u4fc2\u6570\u3092\u63a8\u5b9a\u3059\u308b\u3002\u3057\u304b\u3057\u3001\u5916\u308c\u5024\u304c\u5b58\u5728\u3059\u308b\u3068\u3001\u305d\u306e\u5927\u304d\u306a\u6b8b\u5dee\u304c\u4e8c\u4e57\u3055\u308c\u308b\u3053\u3068\u3067\u975e\u5e38\u306b\u5927\u304d\u306a\u5024\u3068\u306a\u308a\u3001\u56de\u5e30\u76f4\u7dda\u304c\u5916\u308c\u5024\u306b\u5f37\u304f\u5f15\u304d\u305a\u3089\u308c\u3066\u3057\u307e\u3046\u3002\u3053\u308c\u306f\u3001\u5916\u308c\u5024\u304c\u30e2\u30c7\u30eb\u306b\u904e\u5ea6\u306b\u5f71\u97ff\u3092\u4e0e\u3048\u3001\u672c\u6765\u306e\u30c7\u30fc\u30bf\u50be\u5411\u3092\u6b63\u78ba\u306b\u6349\u3048\u3089\u308c\u306a\u304f\u306a\u308b\u3068\u3044\u3046\u554f\u984c\u3092\u5f15\u304d\u8d77\u3053\u3059\u3002<\/p>\n\n\n\n<p>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306f\u3001\u3053\u306e\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u7de9\u548c\u3059\u308b\u305f\u3081\u306b\u3001\u6b8b\u5dee\u306b\u5bfe\u3059\u308b\u8a55\u4fa1\u65b9\u6cd5\u3092\u5de5\u592b\u3059\u308b\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u5916\u308c\u5024\u306b\u5bfe\u3057\u3066\u5c0f\u3055\u306a\u91cd\u307f\u3092\u4e0e\u3048\u305f\u308a\u3001\u6b8b\u5dee\u304c\u5927\u304d\u304f\u306a\u3063\u3066\u3082\u305d\u306e\u5f71\u97ff\u3092\u904e\u5270\u306b\u8a55\u4fa1\u3057\u306a\u3044\u3088\u3046\u306a\u95a2\u6570\uff08<strong>\u640d\u5931\u95a2\u6570<\/strong>\u307e\u305f\u306f<strong>$\\rho$\u95a2\u6570<\/strong>\uff09\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3001\u3088\u308a\u9811\u5065\u306a\uff08\u30ed\u30d0\u30b9\u30c8\u306a\uff09\u63a8\u5b9a\u3092\u53ef\u80fd\u306b\u3059\u308b\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">M\u63a8\u5b9a\uff08M-estimation\uff09<\/h3>\n\n\n\n<p>M\u63a8\u5b9a\u306f\u3001\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306e\u57fa\u672c\u7684\u306a\u624b\u6cd5\u306e\u4e00\u3064\u3067\u3001\u300c<strong>\u6700\u5927\u5c24\u5ea6\u578b\u63a8\u5b9a\uff08Maximum-likelihood type estimation\uff09<\/strong>\u300d\u306e\u7565\u3067\u3042\u308b\u3002\u6700\u5c0f\u4e8c\u4e57\u6cd5\u304c\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u3092\u6700\u5c0f\u5316\u3059\u308b\u306e\u306b\u5bfe\u3057\u3001M\u63a8\u5b9a\u306f\u6b8b\u5dee\u306e\u4f55\u3089\u304b\u306e\u95a2\u6570 $\\rho(\\cdot)$ \u306e\u548c\u3092\u6700\u5c0f\u5316\u3059\u308b\u3002\u3053\u306e $\\rho(\\cdot)$ \u95a2\u6570\u306f\u3001\u6b8b\u5dee\u304c\u5927\u304d\u304f\u306a\u308b\u306b\u3064\u308c\u3066\u305d\u306e\u5f71\u97ff\u3092\u6291\u5236\u3059\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u308b\u3002<\/p>\n\n\n\n<p><strong>\u4ed5\u7d44\u307f:<\/strong><\/p>\n\n\n\n<p>M\u63a8\u5b9a\u306f\u3001\u4e00\u822c\u7684\u306b<strong>\u53cd\u5fa9\u91cd\u307f\u4ed8\u304d\u6700\u5c0f\u4e8c\u4e57\u6cd5\uff08IRLS: Iteratively Reweighted Least Squares\uff09<\/strong>\u3092\u7528\u3044\u3066\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u521d\u671f\u5024\u306e\u8a2d\u5b9a:<\/strong> \u307e\u305a\u3001\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306a\u3069\u3067\u56de\u5e30\u4fc2\u6570\u306e\u521d\u671f\u5024\u3092\u63a8\u5b9a\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u6b8b\u5dee\u306e\u8a08\u7b97:<\/strong> \u73fe\u5728\u306e\u4fc2\u6570\u3092\u7528\u3044\u3066\u3001\u5404\u30c7\u30fc\u30bf\u70b9\u306b\u304a\u3051\u308b\u6b8b\u5dee\uff08\u89b3\u6e2c\u5024\u3068\u4e88\u6e2c\u5024\u306e\u5dee\uff09\u3092\u8a08\u7b97\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u91cd\u307f\u306e\u8a08\u7b97:<\/strong> \u5404\u6b8b\u5dee\u306b\u5bfe\u3057\u3066\u3001\u91cd\u307f\u95a2\u6570 $w(\\cdot)$ \u3092\u7528\u3044\u3066\u91cd\u307f\u3092\u8a08\u7b97\u3059\u308b\u3002\u3053\u306e\u91cd\u307f\u95a2\u6570\u306f\u3001\u6b8b\u5dee\u304c\u5c0f\u3055\u3044\u30c7\u30fc\u30bf\u70b9\u306b\u306f\u5927\u304d\u306a\u91cd\u307f\u3092\u3001\u6b8b\u5dee\u304c\u5927\u304d\u3044\uff08\u5916\u308c\u5024\u306e\u53ef\u80fd\u6027\u304c\u9ad8\u3044\uff09\u30c7\u30fc\u30bf\u70b9\u306b\u306f\u5c0f\u3055\u306a\u91cd\u307f\u3092\u4e0e\u3048\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u308b\u3002\n<ul class=\"wp-block-list\">\n<li>$w(e_i) = \\frac{\\psi(e_i)}{e_i}$ \uff08$e_i$\u306f\u6b8b\u5dee\u3001$\\psi(\\cdot)$\u306f\u5f71\u97ff\u95a2\u6570\u3067 $\\rho(\\cdot)$ \u306e\u5fae\u5206\uff09<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u91cd\u307f\u4ed8\u304d\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306e\u9069\u7528:<\/strong> \u8a08\u7b97\u3055\u308c\u305f\u91cd\u307f\u3092\u7528\u3044\u3066\u3001\u91cd\u307f\u4ed8\u304d\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u5b9f\u884c\u3057\u3001\u65b0\u3057\u3044\u56de\u5e30\u4fc2\u6570\u3092\u63a8\u5b9a\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u53ce\u675f\u5224\u5b9a:<\/strong> \u65b0\u3057\u3044\u4fc2\u6570\u304c\u524d\u306e\u53cd\u5fa9\u306e\u4fc2\u6570\u3068\u5341\u5206\u306b\u8fd1\u304f\u306a\u3063\u305f\u5834\u5408\uff08\u53ce\u675f\u57fa\u6e96\u3092\u6e80\u305f\u3057\u305f\u5834\u5408\uff09\u306f\u8a08\u7b97\u3092\u7d42\u4e86\u3059\u308b\u3002\u305d\u3046\u3067\u306a\u3051\u308c\u3070\u3001\u624b\u98062\uff08\u6b8b\u5dee\u306e\u8a08\u7b97\uff09\u306b\u623b\u308a\u3001\u6b8b\u5dee\u3001\u91cd\u307f\u3001\u4fc2\u6570\u306e\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3059\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>\u4ee3\u8868\u7684\u306a\u640d\u5931\u95a2\u6570\uff08$\\rho$\u95a2\u6570\uff09\u3068\u91cd\u307f\u95a2\u6570:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Huber\u95a2\u6570\uff08Huber Loss\uff09:<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u6b8b\u5dee\u304c\u5c0f\u3055\u3044\u5834\u5408\u306f\u4e8c\u4e57\u548c\u3068\u540c\u3058\u3088\u3046\u306b\u632f\u308b\u821e\u3046\u304c\u3001\u3042\u308b\u95be\u5024\uff08$k$\uff09\u3092\u8d85\u3048\u308b\u3068\u6b8b\u5dee\u306e\u7d76\u5bfe\u5024\u306b\u6bd4\u4f8b\u3059\u308b\u3088\u3046\u306b\u306a\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u6975\u7aef\u306a\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u7dda\u5f62\u306b\u6291\u5236\u3059\u308b\u3002<\/li>\n\n\n\n<li>$\\rho(e) = \\begin{cases} \\frac{1}{2}e^2 &amp; \\text{if } |e| \\le k \\\\ k|e| &#8211; \\frac{1}{2}k^2 &amp; \\text{if } |e| > k \\end{cases}$<\/li>\n\n\n\n<li>\u91cd\u307f\u95a2\u6570\u306f\u3001\u95be\u5024\u5185\u3067\u306f\u4e00\u5b9a\u3001\u95be\u5024\u5916\u3067\u306f\u6b8b\u5dee\u306e\u7d76\u5bfe\u5024\u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Tukey&#8217;s Biweight\u95a2\u6570\uff08Tukey&#8217;s Biweight Loss\uff09:<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u6b8b\u5dee\u304c\u5c0f\u3055\u3044\u5834\u5408\u306f\u4e8c\u4e57\u548c\u306e\u3088\u3046\u306b\u632f\u308b\u821e\u3046\u304c\u3001\u3042\u308b\u95be\u5024\uff08$c$\uff09\u3092\u8d85\u3048\u308b\u3068\u6b8b\u5dee\u306e\u5f71\u97ff\u3092\u5b8c\u5168\u306b\u7121\u8996\uff08\u91cd\u307f\u30920\uff09\u3059\u308b\u3088\u3046\u306b\u306a\u308b\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u6975\u7aef\u306a\u5916\u308c\u5024\u3092\u5f37\u304f\u6392\u9664\u3067\u304d\u308b\u3002<\/li>\n\n\n\n<li>$\\rho(e) = \\begin{cases} \\frac{c^2}{6}\\left[1 &#8211; \\left(1 &#8211; \\left(\\frac{e}{c}\\right)^2\\right)^3\\right] &amp; \\text{if } |e| \\le c \\\\\\frac{c^2}{6} &amp; \\text{if } |e| > c \\end{cases}$<\/li>\n\n\n\n<li>\u91cd\u307f\u95a2\u6570\u306f\u3001\u95be\u5024\u5916\u30670\u3068\u306a\u308b\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p><strong>\u5229\u70b9:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u52b9\u679c\u7684\u306b\u8efd\u6e1b\u3067\u304d\u308b\u3002<\/li>\n\n\n\n<li>\u8a08\u7b97\u304c\u6bd4\u8f03\u7684\u5bb9\u6613\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>\u6b20\u70b9:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u300c\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u300d\uff08\u30c7\u30fc\u30bf\u5168\u4f53\u306e\u3046\u3061\u3001\u3069\u306e\u304f\u3089\u3044\u306e\u5272\u5408\u306e\u5916\u308c\u5024\u307e\u3067\u63a8\u5b9a\u304c\u6709\u52b9\u304b\u3092\u793a\u3059\u6307\u6a19\uff09\u304c\u4f4e\u3044\u5834\u5408\u304c\u3042\u308b\uff08\u6700\u5927\u306750%\uff09\u3002<\/li>\n\n\n\n<li>\u8907\u6570\u306e\u5916\u308c\u5024\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u306b\u3001\u305d\u306e\u691c\u51fa\u304c\u96e3\u3057\u3044\u5834\u5408\u304c\u3042\u308b\uff08<strong>\u30de\u30b9\u30af\u52b9\u679c<\/strong>\uff09\u3002<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">S\u63a8\u5b9a\uff08S-estimation\uff09<\/h3>\n\n\n\n<p>S\u63a8\u5b9a\u306f\u3001\u30ed\u30d0\u30b9\u30c8\u306a<strong>\u5c3a\u5ea6\uff08\u30b9\u30b1\u30fc\u30eb\uff09\u63a8\u5b9a<\/strong>\u306b\u57fa\u3065\u3044\u305f\u63a8\u5b9a\u6cd5\u3067\u3042\u308b\u3002\u56de\u5e30\u30e2\u30c7\u30eb\u306e\u6b8b\u5dee\u306e\u5c3a\u5ea6\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u4fc2\u6570\u3092\u63a8\u5b9a\u3059\u308b\u3002\u3053\u3053\u3067\u3044\u3046\u5c3a\u5ea6\u3068\u306f\u3001\u6b8b\u5dee\u306e\u3070\u3089\u3064\u304d\u306e\u5ea6\u5408\u3044\u3092\u793a\u3059\u3082\u306e\u3067\u3001\u6a19\u6e96\u504f\u5dee\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u308b\u3002S\u63a8\u5b9a\u306fM\u63a8\u5b9a\u3088\u308a\u3082\u9ad8\u3044\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u3092\u6301\u3064\u3053\u3068\u304c\u7279\u5fb4\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<p><strong>\u4ed5\u7d44\u307f:<\/strong><\/p>\n\n\n\n<p>S\u63a8\u5b9a\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u6b8b\u5dee\u306e\u5c3a\u5ea6 $\\hat{\\sigma}$ \u3092\u6700\u5c0f\u5316\u3059\u308b\u56de\u5e30\u4fc2\u6570 $\\hat{\\beta}$ \u3092\u898b\u3064\u3051\u308b\u3053\u3068\u3067\u884c\u308f\u308c\u308b\u3002<br>$$\\min_{\\hat{\\beta}} \\hat{\\sigma}(\\mathbf{e}) \\quad \\text{s.t.} \\quad \\frac{1}{n}\\sum_{i=1}^n \\rho\\left(\\frac{e_i}{\\hat{\\sigma}}\\right) = K$$<br>\u3053\u3053\u3067\u3001$e_i$ \u306f\u6b8b\u5dee\u3001$K$ \u306f\u5b9a\u6570\uff08\u591a\u304f\u306e\u5834\u5408 $E[\\rho(Z)]$\u3001\u305f\u3060\u3057 $Z$ \u306f\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\uff09\u3001$\\rho(\\cdot)$ \u306f\u975e\u6e1b\u5c11\u95a2\u6570\u3067\u3001\u5916\u308c\u5024\u306b\u5bfe\u3057\u3066\u5f71\u97ff\u3092\u6291\u5236\u3059\u308b\u3088\u3046\u306a\u95a2\u6570\uff08\u4f8b\uff1aTukey&#8217;s Biweight\uff09\u304c\u7528\u3044\u3089\u308c\u308b\u3002<\/p>\n\n\n\n<p><strong>\u5229\u70b9:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u9ad8\u3044\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u3092\u6301\u3064\uff08\u6700\u592750%\uff09\u3002\u3053\u308c\u306f\u3001\u30c7\u30fc\u30bf\u306e\u534a\u5206\u8fd1\u304f\u304c\u5916\u308c\u5024\u3067\u3042\u3063\u3066\u3082\u3001\u63a8\u5b9a\u304c\u6709\u52b9\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002<\/li>\n\n\n\n<li>\u9ad8\u6b21\u5143\u30c7\u30fc\u30bf\u306b\u5bfe\u3057\u3066\u3082\u6bd4\u8f03\u7684\u5b89\u5b9a\u3057\u305f\u6027\u80fd\u3092\u793a\u3059\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>\u6b20\u70b9:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>M\u63a8\u5b9a\u306b\u6bd4\u3079\u3066\u8a08\u7b97\u30b3\u30b9\u30c8\u304c\u9ad8\u3044\u3002<\/li>\n\n\n\n<li>\u7d71\u8a08\u7684\u52b9\u7387\u304cM\u63a8\u5b9a\u3088\u308a\u52a3\u308b\u3053\u3068\u304c\u3042\u308b\u3002<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">MM\u63a8\u5b9a\uff08MM-estimation\uff09<\/h3>\n\n\n\n<p>MM\u63a8\u5b9a\u306f\u3001M\u63a8\u5b9a\u3068S\u63a8\u5b9a\u306e\u5229\u70b9\u3092\u7d44\u307f\u5408\u308f\u305b\u305f\u624b\u6cd5\u3067\u3001<strong>\u9ad8\u3044\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9<\/strong>\u3068<strong>\u9ad8\u3044\u7d71\u8a08\u7684\u52b9\u7387<\/strong>\u3092\u4e21\u7acb\u3055\u305b\u308b\u3053\u3068\u3092\u76ee\u6307\u3059\u3002<\/p>\n\n\n\n<p><strong>\u4ed5\u7d44\u307f:<\/strong><\/p>\n\n\n\n<p>MM\u63a8\u5b9a\u306f3\u3064\u306e\u30b9\u30c6\u30c3\u30d7\u3067\u69cb\u6210\u3055\u308c\u308b\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u521d\u671f\u306eS\u63a8\u5b9a:<\/strong> \u307e\u305a\u3001\u9ad8\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u3092\u6301\u3064S\u63a8\u5b9a\u3092\u7528\u3044\u3066\u3001\u56de\u5e30\u4fc2\u6570\u306e\u30ed\u30d0\u30b9\u30c8\u306a\u521d\u671f\u63a8\u5b9a\u5024\u3068\u3001\u6b8b\u5dee\u306e\u30ed\u30d0\u30b9\u30c8\u306a\u5c3a\u5ea6\uff08\u30b9\u30b1\u30fc\u30eb\uff09\u3092\u6c42\u3081\u308b\u3002\u3053\u306e\u30b9\u30c6\u30c3\u30d7\u3067\u3001\u5916\u308c\u5024\u306b\u5f37\u3044\u521d\u671f\u5024\u3092\u5f97\u308b\u3053\u3068\u304c\u76ee\u7684\u3067\u3042\u308b\u3002<\/li>\n\n\n\n<li><strong>M\u63a8\u5b9a:<\/strong> \u30b9\u30c6\u30c3\u30d71\u3067\u5f97\u3089\u308c\u305f\u30ed\u30d0\u30b9\u30c8\u306a\u5c3a\u5ea6\uff08\u30b9\u30b1\u30fc\u30eb\uff09\u3068\u3001\u3042\u308b\u640d\u5931\u95a2\u6570\uff08\u901a\u5e38\u306fHuber\u95a2\u6570\u306a\u3069\uff09\u3092\u7528\u3044\u3066\u3001M\u63a8\u5b9a\u3092\u884c\u3046\u3002\u3053\u306e\u30b9\u30c6\u30c3\u30d7\u3067\u3001\u63a8\u5b9a\u306e\u52b9\u7387\u6027\u3092\u5411\u4e0a\u3055\u305b\u308b\u3002<\/li>\n\n\n\n<li><strong>\u6700\u7d42\u7684\u306aM\u63a8\u5b9a:<\/strong> \u30b9\u30c6\u30c3\u30d72\u3067\u5f97\u3089\u308c\u305f\u4fc2\u6570\u3068\u3001\u5225\u306e\u640d\u5931\u95a2\u6570\uff08\u901a\u5e38\u306fTukey&#8217;s Biweight\u306a\u3069\u3001\u3088\u308a\u5f37\u529b\u306a\u5f71\u97ff\u6291\u5236\u529b\u3092\u6301\u3064\u3082\u306e\uff09\u3092\u7528\u3044\u3066\u3001\u518d\u3073M\u63a8\u5b9a\u3092\u884c\u3046\u3002\u3053\u306e\u30b9\u30c6\u30c3\u30d7\u3067\u3001\u3055\u3089\u306b\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u6392\u9664\u3057\u3001\u6700\u7d42\u7684\u306a\u9ad8\u52b9\u7387\u306a\u63a8\u5b9a\u5024\u3092\u5f97\u308b\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>\u5229\u70b9:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>S\u63a8\u5b9a\u3068\u540c\u69d8\u306b\u9ad8\u3044\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u3092\u6301\u3064\uff08\u6700\u592750%\uff09\u3002<\/li>\n\n\n\n<li>M\u63a8\u5b9a\u3068\u540c\u69d8\u306b\u9ad8\u3044\u7d71\u8a08\u7684\u52b9\u7387\u3092\u6301\u3064\u3002\u3064\u307e\u308a\u3001\u5916\u308c\u5024\u306b\u5f37\u304f\u3001\u304b\u3064\u8aa4\u5dee\u304c\u6b63\u898f\u5206\u5e03\u306b\u5f93\u3046\u7406\u60f3\u7684\u306a\u72b6\u6cc1\u306b\u8fd1\u3044\u5834\u5408\u3067\u3082\u3001\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u8fd1\u3044\u6027\u80fd\u3092\u767a\u63ee\u3059\u308b\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>\u6b20\u70b9:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>M\u63a8\u5b9a\u3084S\u63a8\u5b9a\u306b\u6bd4\u3079\u3066\u8a08\u7b97\u304c\u8907\u96d1\u306b\u306a\u308b\u3002<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u307e\u3068\u3081<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>\u63a8\u5b9a\u6cd5<\/th><th>\u7279\u5fb4<\/th><th>\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9<\/th><th>\u7d71\u8a08\u7684\u52b9\u7387<\/th><\/tr><\/thead><tbody><tr><td><strong>\u6700\u5c0f\u4e8c\u4e57\u6cd5<\/strong><\/td><td>\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u3092\u6700\u5c0f\u5316\u3002\u5916\u308c\u5024\u306b\u975e\u5e38\u306b\u654f\u611f\u3067\u3042\u308b\u3002<\/td><td>0%<\/td><td>\u9ad8\u3044\uff08\u6b63\u898f\u5206\u5e03\u6642\uff09<\/td><\/tr><tr><td><strong>M\u63a8\u5b9a<\/strong><\/td><td>\u6b8b\u5dee\u306e\u7279\u5b9a\u306e\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3002\u5916\u308c\u5024\u306e\u5f71\u97ff\u3092\u8efd\u6e1b\u3059\u308b\u3002<\/td><td>\u4f4e\u3044\uff08\u7d0450%\uff09<\/td><td>\u9ad8\u3044<\/td><\/tr><tr><td><strong>S\u63a8\u5b9a<\/strong><\/td><td>\u6b8b\u5dee\u306e\u5c3a\u5ea6\u3092\u6700\u5c0f\u5316\u3002\u9ad8\u3044\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u3092\u6301\u3064\u3002<\/td><td>\u9ad8\u3044\uff0850%\uff09<\/td><td>\u4f4e\u3044<\/td><\/tr><tr><td><strong>MM\u63a8\u5b9a<\/strong><\/td><td>S\u63a8\u5b9a\u3067\u521d\u671f\u5024\u3092\u6c42\u3081\u3001M\u63a8\u5b9a\u3067\u52b9\u7387\u3092\u9ad8\u3081\u308b\u3002\u9ad8\u30d6\u30ec\u30a4\u30af\u30c0\u30a6\u30f3\u70b9\u3068\u9ad8\u52b9\u7387\u3092\u4e21\u7acb\u3055\u305b\u308b\u3002<\/td><td>\u9ad8\u3044\uff0850%\uff09<\/td><td>\u9ad8\u3044<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u30ed\u30d0\u30b9\u30c8\u63a8\u5b9a\u6cd5\u306f\u3001\u73fe\u5b9f\u306e\u30c7\u30fc\u30bf\u306b\u3088\u304f\u898b\u3089\u308c\u308b\u5916\u308c\u5024\u3084\u975e\u6b63\u898f\u6027\u3068\u3044\u3063\u305f\u554f\u984c\u306b\u5bfe\u51e6\u3059\u308b\u305f\u3081\u306e\u5f37\u529b\u306a\u30c4\u30fc\u30eb\u3067\u3042\u308b\u3002\u7279\u306bMM\u63a8\u5b9a\u306f\u3001\u9ad8\u3044\u9811\u5065\u6027\u3068\u52b9\u7387\u6027\u3092\u517c\u306d\u5099\u3048\u3066\u3044\u308b\u305f\u3081\u3001\u56de\u5e30\u5206\u6790\u3067\u5916\u308c\u5024\u304c\u61f8\u5ff5\u3055\u308c\u308b\u5834\u5408\u306b\u63a8\u5968\u3055\u308c\u308b\u3053\u3068\u304c\u591a\u3044\u624b\u6cd5\u3067\u3042\u308b\u3002\u3069\u306e\u624b\u6cd5\u3092\u9078\u629e\u3059\u308b\u304b\u306f\u3001\u30c7\u30fc\u30bf\u306e\u6027\u8cea\u3084\u5206\u6790\u306e\u76ee\u7684\u306b\u5fdc\u3058\u3066\u614e\u91cd\u306b\u691c\u8a0e\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n\n\n\n\n","protected":false},"excerpt":{"rendered":"<p>\u56de\u5e30\u5206\u6790\u306f\u3001\u5909\u6570\u9593\u306e\u95a2\u4fc2\u6027\u3092\u660e\u3089\u304b\u306b\u3059\u308b\u5f37\u529b\u306a\u7d71\u8a08\u30c4\u30fc\u30eb\u3067\u3042\u308b\u3002\u3057\u304b\u3057\u3001\u30c7\u30fc\u30bf\u306e\u4e2d\u306b\u300c\u5916\u308c\u5024\u300d\u3068\u547c\u3070\u308c\u308b\u7279\u7570\u306a\u5024\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u3001\u901a\u5e38\u306e\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3067\u306f\u5206\u6790\u7d50\u679c\u304c\u5927\u304d\u304f\u6b6a\u3081\u3089\u308c\u3066\u3057\u307e\u3046\u3053\u3068\u304c\u3042\u308b\u3002\u3053\u306e\u3088\u3046\u306a\u554f\u984c\u3092\u89e3\u6c7a\u3057\u3001\u3088\u308a 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